Acta Polytechnica Hungarica
Vol. 9, No. 5, 2012
Double Layer Tensegrity Grids
Tatiana Olejnikova
Department of Applied Mathematics, Civil Engineering Faculty
Technical University of Košice
Vysokoškolská 4, 042 00 Košice, Slovakia
e-mail: tatiana.olejnikova@tuke.sk
Abstract: This paper describes the geometry of a double layer tensegrity grids assembled of
three or four strut prismatic cells. The elementary cells are self-equilibrated and so is their
assembly. The paper shows the creation of a planar grids composed of elementary
equilibrium and grids with single or double curvatures composed of modified equlibrium
shapes.
Keywords: tensegrity system; compression; tension; equilibrium; prismatic cell; grid
structures
1
Introduction
A tensegrity structure is a class of tension structures consisting of tensile members
and compressive members. In civil engineering, the light-weight characteristic of
tensegrity is recognized as a significant advantage for space structures over
conventional structural systems. According to the definition given by Fuller
(1975), a tensegrity structure is a jointed structure consisting of continuous tensile
members (cables) and discontinuous compressive members (struts).
A tensegrity system is established when a set of discontinuous compression
components interacts with a set of continuous tensile components to define
a stable volume in space. However, it needs to be slightly modified, taking into
account the following factors: the components in compression are included inside
the set of components in tension, and the stability of the system is self-equilibrium
stability.
There are three types of tensegrity systems: tensegrity grids, tensegrity
frameworks and tensegrity domes. This article describes the geometric relations of
tensegrity grids assembled of prismatic cells. There are three orbits of members:
horizontal cables, vertical cables and struts. Each node is connected by two
horizontal cables lying in a horizontal plane, one vertical cable and one strut. The
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T. Olejníková
Double Layer Tensegrity Grids
vertical cables and struts connect nodes in different horizontal planes. The
members in each orbit are of equal length. The necessarily even number of nodes
characterises the case of tensegrity systems. If “n” is this number, and if attention
is paid to the spatial case, the minimal value of n is 6.
2
Geometry of the Prismatic Three-Strut Tensegrity
Grids
The geometry of a spatial reticulate system is completely defined by its relational
structure and by the knowledge of the coordinates x(k), y(k), z(k), k = 1,…,n, for
its “n” nodes in reference to a chosen axis system. It is necessary for all internal
elements (struts) to have the same length “s” and for the external elements (cables)
to have the same length “c”.
If the length of the struts is insufficient, the set of envelope elements will not have
a definite shape (the system is then kinematically indeterminate). A first geometry
is defined as a triangular prism (Fig. 1). This system is unstable, and another shape
can be defined (Fig. 2) by relative rotation of the two triangles of cables in the
parallel planes.
6
4
3
6
5
h
5
1
3
r
1
6
2
Figure 1
Triangular prism
4
2
Figure 2
Equilibrium, axonometric and top view
For a given value of the strut length, the totality of the cable net takes a singularly
definite shape, which will be referred to as “null self-stress equilibrium
geometry”. The geometric distance between the nodes corresponds strictly to the
length of manufactured elements. Two geometrical ranges can be identified. If the
relationship between the lengths of the struts and cables is ratio s c  2  1,467
and the relative rotation between triangles is  6 . When the parameter r is
specified, then the length of the struts is s  r 3  2 3 , the length of the all
cables is c  r 3 and the height of the equilibrium is h  r 1 3 .
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Acta Polytechnica Hungarica
2.1
Vol. 9, No. 5, 2012
Bi-Dimensional Assemblies
This section describes examples of bi-dimensional assemblies. Several junction
modes can be used: node on node, node on cable, cable on cable.
In type a) in Figure 3, the junction is operated with a single vertical (bracing)
cable: each of two ends of the strut lies in different horizontal planes, and it will
be used in a plane configuration leading to a double layer grid, in types b) and c),
the junction is operated with horizontal cables.
a)
Junction: node on vertical cable
b)
Junction: node on horizontal cable
c)
Figure 3
Junction modes “node on cable”
A planar double layer tensegrity grid (Fig. 5) is created by using three-struts cells
via a node on cable junction of type a) (Fig. 4).
Figure 4
Three cells junction in mode “node on vertical cable” of type a)
The node coordinates xk , yk , zk  of one cell in the planar grid are in (1):
xk   k 0 r cos k , y k   k 0 r sin k , z k   h1
for k  1,2,3
k   
for k  4,5,6
k  

2
 k - 1 , h1  0
12
3

2
 k - 4 , h1  h  r 1  3
12
3
– 97 –
(1)
T. Olejníková
Double Layer Tensegrity Grids
where the parameter k0  1 , r is a radius of a triangle created by the three
horizontal cables in the cell, h is a height of the cell (Fig. 2).
Figure 5
Planar double layer tensegrity grid by three-struts cells
The node coordinates x1 i, j, k , y1 i, j, k , z1 i, j, k  of all cells in the planar grid
displayed in Figure 5 are
x1 i, j, k   xk   d x  j, y1 i, j, k   yk   d y  j, z1 i, j, k   zk 
d x (j)  vx (i) cos(
j
j
j
j
) - vy (i) sin( ) , d y (j)  vx (i) sin( )  vy (i) cos( )
3
3
3
3
(2)
(3)
where i  1,...,6,j  1,...,6,k  1,...,6, coordinates xk , yk , zk  are expressed in
(1), and
v1 
3r
3r
, v2 
, vx 1  v y 1  0, vx 2  v1 , v y 2  v2 ,
2
4
vx 3 
5
9
v1 , v y 3  v2 , vx 4  2v1 , v y 4  2v2 , vx 5  3v1 ,
4
2
v y 5  3v2 , vx 6 
(4)
9
11
3
v1 , v y 6  v2 , vx 7   v1 , v y 7   8v2 .
4
2
2
By maintaining the principle of elementary self-stressed cells, it is possible to
modify the equilibrium shape so as to generate a double curvature system and
spherical surface with a radius R. The elementary cell must be modified by the
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Acta Polytechnica Hungarica
Vol. 9, No. 5, 2012
parameter k 0 in the equations (1) of the node coordinates of the upper triangles,
for k  4,5,6 , i  1, ...,6, j  1, ...,6 .
The transformation equations of the node coordinates of the lower layer of the
planar grid to the spherical surface with a radius R and centre S(0,0,-R) for
k  1,2,3 , i  1,...,6,j  1,...,6 and parameter k0  1 in equations (1) are:
Px1 i, j, k , y1 i, j, k , z1 i, j, k   Px i, j, k , y  i, j, k , z  i, j, k 
x i, j, k   R cos u i, j, k  cos v i, j, k 
(5)
y  i, j, k   R cos u i, j, k  sin v i, j, k 
z  i, j, k   R sin u i, j, k   R
where
u i, j, k  
 d i, j, k 

, d i, j, k   x12 i, j, k   y12 i, j, k ,
2
R
x i, j, k 
v i, j, k   sgn y1 i, j, k  arccos 1
d i, j, k 
(6)
coordinates x1 i, j, k , y1 i, j, k  are expressed in equations (2). The transformation
equations of the node coordinates of the upper layer of the planar grid to the
spherical surface with a radius R+h and a centre S(0,0,-R) for k  4,5,6 and
Rh
parameter k0 
in equations (1) are
R
x i, j, k   ( R  h) cos u i, j, k  cos v i, j, k 
y i, j, k   ( R  h) cos u i, j, k  sin v i, j, k 
(7)
z  i, j, k   ( R  h) sin u i, j, k   R
where
d i, j, k   x12 i, j, k   y12 i, j, k , u i, j, k  
x i, j, k 
v i, j, k   sgn y1 i, j, k  arccos 1
d i, j, k 
 d i, j, k 

,
2
Rh
(8)
Figure 6 shows the grid composed of three-strut cells transformed to the spherical
surface. Figure 7 illustrates the transformation of the point Pk  to the point
Pi, j, k  , both located in the tangent plane of the spherical surface and its
transformation to the point Pi, j, k  located on the spherical surface. Figure 8
displays a determination of the parameter d i, j, k  and angles ui, j, k  and
vi, j, k  , which are used in equations (8).
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T. Olejníková
Double Layer Tensegrity Grids
Figure 6
Double curvature tensegrity grid system
z
P k 
x
u(i,j,k)
P i, j, k 
P i, j, k 
P i, j, k 
0
P i, j, k 
d(i,j,k)
y
v(i,j,k)
S
Figure 7
k   Pi, j, k   Pi, j, k 
Transform.: P
Figure 8
Determination of d(i,j,k),u(i,j,k),v(i,j,k)
In Figure 9 are displayed the nodes and cables of the triangles of the bottom layer
on the spherical surface. Figure 10 displays the triangles of the bottom layer of the
tensegrity grid located on the tangent plane of the spherical surface, and in
Figure 11 are these triangles placed on the spherical surface together with a
spherical surface.
Figure 9
Nodes and cables of the bottom layer of the spherical grid
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Acta Polytechnica Hungarica
Vol. 9, No. 5, 2012
Figure 10
Triangles on the tangent plane
3
Figure 11
Triangles on the sphere
Geometry of the Prismatic Four-Strut Tensegrity
Grids
Geometry of the four-strut cell is dependent upon only one parameter a (Fig. 11).
When the topology is defined, then the geometry is qualified by the whole set of
coordinates, which is closely related to the self-stress equilibrium. When the
parameter a is specified, then the length of the struts is s  a 20 3 , the length of
the cables of the bottom layer is c1  a 2 , length of the cables of the upper layer
is c2  2a and the height of the equilibrium is h  a 5 3 .
7
8
5
7
4
3
6
8
4
6
h
3
1
2
1
a
5
2
Figure 11
Axonometric, top view and frontal view of the 4-strut cell
Planar double layer tensegrity grid is created by using of four-struts cells by node
on node junction. The node coordinates P k   xk , yk , zk  for k  1,2,3,4
and k  5,6,7,8 of one cell in the planar grid are expressed in equations (9)
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T. Olejníková
Double Layer Tensegrity Grids
 
for k  1,2,3,4 and  k   k - 1  , z k   0,
2 4
P k   xk , y k , z k  

2 a cos  k   a,

for k  5,6,7,8 and  k   k - 4 , z k   a
2

2 a sin k   a, z k 
(9)
1
 2
2
P k   xk , y k , z k    a cos  k   a, a sin k   a, z k 
where parameters a, h are illustrated in Figure 11. A planar double layer tensegrity
grid is created by using 33 four-struts cells by node on node junction. The
coordinates of the nodes in the planar grid for i  1,...,3, j  1,...,3, k  1,...,8 are in
equations (10)
x i, j, k   x k   2a i  1,
y i, j, k   y k   2a  j  1,
(10)
z i, j, k   z k ,
where x k , y k , z k  are coordinates of the nodes of one cell expressed in (9).
Figure 12 shows the planar double layer grid composed of 9 four-strut cells (top
view and frontal view).
Figure 12
Top and frontal view of the planar double layer tensegrity grid of 33 cells
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Acta Polytechnica Hungarica
Vol. 9, No. 5, 2012
By maintaining the principle of elementary self-stressed cells it is possible to
modify the equilibrium shape so as to generate a single curvature tensegrity grid
system (on the cylindrical surface). Then the node coordinates are expressed in
transformation equations (11), where the nodes located in the plane are
transformated to the nodes located on the cylindrical surfaces with radii R and
R+h, P k   P i, j, k  , i  1,...,7, j  1,...,7
x i, j, k   x k  cos  i  z k   R  sin  i ,
y i, j, k   y k   2a  j  1,
(11)
z i, j, k   x k  sin  i  z k   R  cos  i  R,
where parameter a for nodes
Pk , for k  6,8 in (10) is modified on
b  a1  h R  illustrated in Figure 14, parameter i  6  i  1 2 , where
  arctan a R , R is radius of the curvature of the single curvature system.
Figure 13
Single curvature double layer tensegrity grid of 77 four-strut cells
In Figure 14 is displayed a 1 modified four-strut cell and in Figure 15 a
2 modified one. Figure 13 displays a single curvature double layer tensegrity grid
containing 77 1 modified four-strut cells.
The node coordinates of the grid with double curvature (created surface with two
curvatures R1 and R2) are
x i, j, k   x k  cos  i  z k   R1  sin  i ,
y i, j, k   y k ,
(12)
z  i, j, k   x k  sin  i  z k   R1  cos   R1 ,
– 103 –
T. Olejníková
Double Layer Tensegrity Grids
7
7
4
4
3
3
b2
8
6
8
6
b
b1
1
5
a
1
2
2
5
Figure 14
1 modified four-strut cell
Figure 15
2 modified four-strut cell
xi, j, k   xi, j, k 
y i, j, k   yi, j, k  cos  j  z i, j, k   R2  sin  j ,
(13)
z i, j, k   yi, j, k sin  j  z i, j, k   R2  cos  j  R2 ,
where parameter a for nodes
Pk , if k  6,8 in (10) is modified on
b1  a 1  h R1  , and for nodes Pk , if k  5,7 on b2  a 1  h R2  ,
  arctan a r1  ,   arctan a r2  , i  6  i  1 2 , i  6   j  1 2 ,
where r1  sgn1 R1 and r2  sgn 2 R2 are the radii of the curvatures of the double
curvature system with its orientation determined by the parameter sgn1, 2  1 .
In Figure 16 is displayed a double curvature double layer tensegrity grid
containing 77 2 modified four-strut cells, where the radii of curvature are the
same size and the same orientation r1  r2 . This grid has the form of a spherical
surface.
Figure 16
Double curvature double layer tensegrity grid 77, r1
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Acta Polytechnica Hungarica
Vol. 9, No. 5, 2012
In Figure 17 the tensegrity grid with the same radii but with opposite orientation
r1  r2 has the form of the translational surface created by the translation of the
circle with radius R1 along the circle with radius R2 . In Figure 18 the tensegrity
grid with different radii and opposite orientation r1  4 r2 has the form of the
translational surface created by the translation of the circle with radius R1 along
the circle with radius R2 .
Figure 17
Double curvature double layer tensegrity grid with radii r1
Figure 18
Double curvature double layer tensegrity grid with radii r1
– 105 –
 r2
 4 r2
T. Olejníková
Double Layer Tensegrity Grids
Conclusions
This paper aimed to provide some examples of planar or non-planar prismatic
tensegrity grid systems. Some are only geometrical studies without equilibrium
considerations. It is not easy to define tensegrity systems. It could be claimed that
everything in the universe is tensegrity with properties related to the continuum of
tensioned components. Tensegrity structures are the most recent addition to the
array of systems available to designers. The concept itself is about eighty years
old and it came not from within the construction industry but from the world of
arts. Although its basic building blocks are very simple – a compression element
and a tension element – the manner in which they are assembled in a complete,
stable system is by no means obvious.
The idea was adopted into architecture in 1960 when Maciej Gintowt and Maciej
Krasiński, architects of Spodek, a venue in Katowice in Poland, designed it as one
of the first major structures to employ the principle of tensegrity. The roof uses an
inclined surface held in check by a system of cables holding up its circumference.
Another example of a practical implementation of the tensegrity system is Seoul
Olympic Gymnastics Arena designed by David Geiger in 1980 for the 1988
Summer Olympics. The Georgia Dome, which was built for the 1996 Summer
Olympics, is a large tensegrity structure of similar design to the aforementioned
Gymnastics Hall.
Acknowledgement
This work was supported by the Slovak Grant Agency of Ministry of Education of
the Slovak Republic within the project VEGA No. 1/0321/12 “Theoretical and
experimental analysis of adaptive cable and tensegrity systems under static and
dynamic stress considering the effect of wind and seismic”.
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